I think the most suitable agile methodology for this scenario is Scrum, since it is a management methodology and can be applied to a wide range of situations (including scenarios were there are not developers, for example).
I would recomend starting with a book the describe as the basics and then you can personalize scrum the way you find more suitable:
Schwaber, K., & Beedle, M. (2002). Agile software development with Scrum (Vol. 1). Upper Saddle River: Prentice Hall.
You can find good starting materials online, provided by Scrum Alliance ans The Scrum Guide:
I think you can get a lot of benefits if you use Scrum and Kanban boards and its daily meetings. The other suggestion is that if the project is bigger and complicated, you can combine Agile and Scrum with some methods such as UML (RUP). RUP provides a structured and methodological way of development for large scaled projects (Kruchten, 2007); while as a process model, it provides well-defined step by step guidelines for software development.
Scrum and Agile are fast in terms of development and support communication and iteration to a high extent.
By the way, there are lots of free software you can test before buying a professional one (such as Jira) for Scrum. If using Jira, there is a separated part for developers and business users to collaborate in Kanban board.